On 1/20/11 at 6:30 AM, nbbienia at cyf-kr.edu.pl (Leslaw Bieniasz)
wrote:
>On Wed, 19 Jan 2011, Bill Rowe wrote:
>>On 1/18/11 at 5:48 AM, nbbienia at cyf-kr.edu.pl (Leslaw Bieniasz)
>>wrote:
>>>I need to determine minimax polynomial approximations to a certain
>>>function computed using MATHEMATICA. Unfortunately it is not
>>>possible to calculate exact derivatives of the function. Is there
>>>any way to use the MiniMaxApproximation[] algorithm with
>>>numerically approximated derivatives? I would appreciate an
>>>example.
>>A minmax approximation can be efficiently computed as a Chebyshev
>>series. You don't need to compute the derivative to get the
>>coefficients for a Chebyshev series. All you need do is sample the
>>function at appropriate points. Then the needed coefficients can be
>>computed using a discrete cosine transform.
>>
>>See the applications section of ref/FourierDCT for details of how
>>to sample the function correctly and use FourierDCT to compute the
>>needed coefficients.
>Sorry, I don't grasp that. Minimax polynomial results from the
>minimisation of the norm maximum of the difference between the
>polynomial and a given function. Therefore, some optimisation
>procedure must be used (often the Remez algorithm is used for this
>purpose).
Yes and no. If you want the optimal minimax polynomial, then yes
you will need to use an optimization algorithm. By optimal, I
mean the minimiax polynomial that has the smallest absolute
error for a given polynomial degree. But if you are willing to
settle for a minimax polynomial that is nearly optimal, then the
Chebyshev series will do and can be computed without computing
the derivatives.
See <http://en.wikipedia.org/wiki/Approximation_theory> for more details.
Also, a very good starting point for the Remez algorithm is a
Chebyshev series.
See <http://en.wikipedia.org/wiki/Remez_algorithm>